Cops don’t even have the ability to enforce laws of the United States outside their jurisdiction. That New York cop you mentioned couldn’t arrest/detain/engage a criminal if they were on vacation in LA. Depending on the place some departments might allow it, but just as often they get annoyed.

There used to be a retired sheriff in my neighborhood growing up who maintained his peace officer license. He had lights and a siren on his personal vehicle and would pull people over and ticket them. The local court though would just dismiss everything he issued.

Abroad it would depend heavily on the location, but it seems unlikely many countries would tolerate a foreign law enforcement officer doing anything related to police work. There are all sorts of treaties and agreements about that sort of thing so likely a vacationing cop would create an international incident if they tried anything. I’m sure in some situations like murder or sexual assault they might get a pass, but it seems unlikely.